Can an AI make a constructed (natural) language?
could an AI make construct it's own natural language, with words,
  conjugations and grammar rules?


Sure. This might be helpful: Simulated Evolution of Language: a Review of the Field


  Basically, a language that humans could use to speak to each other.
  (Preferably to communicate abstract, high-level concepts.)


I'm not sure how useful such a machine adapted language would be to human speech. I suspect not very. Perhaps it could be useful as a kind of "common byte-code format" to translate between multiple human languages... But English kind of already serves in that role. Doing so would probably be an academic exercise.


  Could it be based on existing natural languages or would it have few
  connections to existing natural languages?


You could probably generate languages in either direction. Languages not linked to human-natural languages will probably take shapes that reflect the problem spaces they work with. For instance, if this is an ant simulation, the generated words will probably reflect states related to food, energy, other ants, etc.


  Could it design a language that's easier to learn than existing
  languages (even Esperanto)?


Easier for a machine? Definitely. Easier for a human? Probably not. Our brains are somewhat adapted to the languages we use. And the languages we use are somewhat adapted to our brains.

What is your goal? To create a language that is easier to use for humans than existing human languages?

If your intention is to build a "universal language" that could be "the most efficient" for machines, humans and aliens - no such thing can exist. The space of all possible machines is infinite and therefor limits our ability to define communicative abstractions that have utility across all contexts.

If we make lots of assumptions about machines, like they have intentions, they exist in 3 dimensions, they differentiate between temporally linked events, they have eye balls, a need to consume foods and liquids, a need to carry the food from place to place, etc... Then yes, common communicative abstractions may have utility across the set of those kinds of machines. But then we're no longer dealing with the general case, but one much more specific.

These links also seem interesting and somewhat related:


Grammar Induction and Genetic Algorithms: An Overview. [pdf]
A Model of Children's Acquisition of Grammatical Word Categories Using an Adaptation and Selection Algorithm [pdf]
The processing of verbs and nouns in neural networks: insights from synthetic brain imaging